Answer:
g and f are inverse functions because g(f(x)) = f(g(x)) = x
Step-by-step explanation:
Let's start by finding f(g(x)) and g(f(x)). As we discussed in another question, to find a composite function, apply the outer function to whatever the inner function evaluates to. We can start with f(g(x)):
[tex]f(g(x))=f(\frac{x+6}{7})=7(\frac{x+6}{7} )-6=x+6-6=x[/tex]
Now, let's find g(f(x)):
[tex]g(f(x))=g(7x-6)=\frac{(7x-6)+6}{7} =\frac{7x}{7}=x[/tex]
There is a property that says that if f(g(x)) = g(f(x)) = x, the two functions are inverse. This suggests that g and f are inverse functions. We can verify this by taking one function, switching y and x, and then solving for y. If we complete this process and find that we get the OTHER function, it means the two functions are inverse. Let's try that:
[tex]f(x)=y=7x-6[/tex]
Swap x and y:
[tex]x=7y-6[/tex]
Solve for y:
[tex]x+6=7y\\y=\frac{x+6}{7}[/tex]
Notice that we got g, which means f and g are inverse.
